By "throttling", blue_cowdawg did not mean blocking, but rather limiting the rate at which new messages can be sent

You might consider adding a sleep(1) in your loop.

Also, maybe you should consider using a list service. You send your message to the service, then it forwards it to the list subscribers.

Since you are already using Google, Google Groups is one list service to consider. In my (limited) experience of subscribing to Google Groups, subscribers don't have to have a Google account to do so (assuming it hasn't changed since the last time I used a non-Google account with a Google Group).

Thanks to all for the help. I seem to have things under control now. You guys were right that Google is apparently rejecting the messages, but it seems to be after about 90, which is sufficient for my purposes. $mail->send does return a single value in a list variable which appears to be 1 for success and -1 for failure, so looking at that lets me know where the block begins.

Good that you understand what's happening

I still recommend putting a delay in your loop to slow the process down so Google is less likely to throttle your outgoing.